An Invitation to Variational Methods in Differential Equations David G. Costa Birkhäuser (2007), 138 pp., Price: €39.90 ISBN 978-0-8176-4535-9.
Reviewer: David A. W. Barton Bristol Centre for Applied Nonlinear Mathematics University of Bristol Bristol, England, U.K.

This book is intended to be an introduction to variational methods for ODEs and PDEs. The title is slightly misleading, since really it is an introduction to critical point theory more than anything else. Although this book is short (just over 120 pages) it covers a lot of material, which is both a blessing and a curse. The reader is left with a flavour of the subject as the book lightly touches upon many of the important theorems in the area, such as the Mountain Pass theorem and Rabinowitz's saddle-point theorem, but it is easy to lose their significance in the brevity of treatment.

The book is broken into 11 chapters (plus appendix), each of which, with the exception of the introduction, focuses on a particular theorem. The structure of each chapter is simple but effective; there is a short descriptive introduction to the theorem being covered, followed by the actual theorem itself and a worked example (or two) of its use on a particular problem. Each chapter ends with a reasonably large (given the size of the chapters) selection of exercises for the reader.

The first chapter gives the reader a gentle introduction to the usefulness of variational methods, starting with five elementary ODE boundary value problems and showing how variational methods can be used to find their solutions. There is a sudden change of pace as you hit the second chapter; the author clearly expects the reader to have a reasonable grounding in functional analysis and dives straight into the technical details. The initial onrush of detail is relatively easy to deal with as there are no demanding concepts introduced. However, the subsequent worked examples are considerably more taxing to follow.

In chapters 3, 4 and 5 the book settles into its pattern of theorem, examples, exercises and covers the deformation theorem, the mountain-pass theorem and a saddle-point theorem (including a brief mention of topological degree). The occasional diagrams throughout these chapters make a big difference to those of us who are more geometrically minded. Unfortunately, the speed at which these theorems are covered makes it easy to brush them aside without thinking—to understand the full worth of these theorems it is really necessary to attempt some of the exercises at the end of each chapter. The author expects you to work for the understanding!

Chapters 6-11 cover a range of topics including critical points under constraints, duality and several chapters on symmetries and lack of compactness. Again, the coverage of these topics is at best brief, but on the positive side each chapter is (mostly) self-contained and so it is easy to dip into the topics of most interest to you. Should a deeper treatment of a topic be needed, suitable references are provided; however, the list of references provided is not that extensive, but it does contain the principal text books in the field.

From my point of view, the chapter on the mountain-pass theorem was the most enlightening. It starts off with a single sentence introduction to minimax problems, succinct but sufficient, before presenting three examples to give the reader an idea of their applicability. The first two examples chosen are on the minimax characterisation technique of eigenvalues of a real symmetric matrix (Fischer, 1905) and its generalisation to symmetric operators on Hilbert spaces. The third example deals with a topological analogue to minimax schemes. As they stand the examples are of limited use; they are simply too brief. However, as just a flavour of the subject they work well. In a similarly brief manner the mountain-pass theorem (with proof) is presented in a total of two pages. Fortunately, the author has a very clear writing style and it is easy to follow. The strength of this book shines through in the penultimate section of this chapter: the worked examples. The author uses the mountain-pass theorem to prove the existence of non-trivial solutions to $-\\Delta u=u^3$ with zero boundary conditions and carefully works through each point of the proof. The second example is then a generalisation to arbitrary right-hand-sides (subject to some constraints) worked out in similar detail.

Overall, this is a well-written book on the variational methods, but I think that its use on its own as an introductory text is limited. The author clearly loves and knows the subject area and it is very good at providing an overview to the area; however, unless you have some prior knowledge, the treatment is far too fast and concise to enable the reader to understand the subtleties of the theorems presented. There is, though, potential to learn a great deal from the examples and exercises at the end of each chapter. As such, a good guided reading course for graduate students could be made from this book, covering one chapter per session (or two).